The graft transformation conjecture for the distance spectral radius

Let $G_c(s,t)$ be the connected uniform hypergraphs obtained from the graft transformation described in the paper, with distinguished vertex $r_s$ in the branch corresponding to $s$. Assume

$$|E(G_{r_s})|>c$$

and

$$d_{G_{r_s}}(r_s)\geq c.$$

Graft transformation conjecture. For $s\geq t\geq 2$,

$$\rho(G_c(s+1,t-1))>\rho(G_c(s,t)).$$

This conjecture predicts that transferring one unit from the shorter branch to the longer branch strictly increases the distance spectral radius under the stated size and degree conditions. The supplied context indicates that it is motivated by graft transformations and Lemma 7, but does not establish the claim or provide a resolution.